Recognition by prime graph of the almost simple group PGL(2, 25)

Subject Areas : History and biography

A. Mahmoudifar ¹

1 - Department of Mathematics, Tehran-North Branch, Islamic Azad University, Tehran, Iran

Received: 2015-12-24 Accepted : 2016-05-20 Published : 2016-06-01

Keywords: prime graph, linear group, Almost simple group, element order, Frobenius group,

Abstract :

Throughout this paper, every groups are finite. The prime graph of a group $G$ is denotedby $\Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $\Gamma(H) = \Gamma(G)$, we conclude that $G\cong H$. Until now, it is proved that if $k$ is an odd numberand $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ iseven, the recognition by prime graph of $PGL(2,p^k)$, where $p$ is an odd prime number, is anopen problem. In this paper, we generalize this result and we prove that the almost simplegroup $PGL(2,25)$ is recognizable by prime graph.

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